Reduction of Warpage of Multilayered Substrate or Package

1. A computer implemented method of performing a calculation associated with controlling warpage in a multilayered substrate, the method comprising the steps of: (1) discretizing a two-dimensional surface of the multilayered substrate into finite planar elements, wherein the multilayered substrate has an upper wiring layer and a lower wiring layer, wherein the upper and lower wiring layers are made of at least two different materials having different elastic moduli, and wherein the materials are distributed two-dimensionally with different occupancies; (2) calculating a difference in coefficient of thermal expansion of the two-dimensional surface based on composite properties determined from two elastic moduli, two occupancies, and difference between the two occupancies obtained for each finite planar element; (3) calculating the warpage of the multilayered substrate based on the calculated composite properties of the two-dimensional surface according to a finite element method; (4) calculating a maximum length scale that corresponds to an allowable warpage; (5) extracting an element involved with a pattern having a length scale greater than a maximum length scale by digital filtering of the difference in coefficient of thermal expansion to produce an extracted element; (6) reducing the difference in coefficient of thermal expansion by adjusting a mixing ratio of the two different occupancies for the extracted element involved with a pattern having a length scale greater than the maximum length scale to produce an adjusted mixing ratio; (7) calculating the warpage of the multilayered substrate based on the adjusted mixing ratio of the two different occupancies according to the finite element method; and (8) repeating steps (5), (6) and (7) until the calculated warpage falls below a required warpage.

2. The method according to claim 1 , wherein the digital filtering of step (5) is performed after a fast Fourier transform.

3. The method according to claim 1 , wherein an inverse fast Fourier transform is performed after the digital filtering.

4. The method according to claim 1 , wherein step (6) of adjusting a mixing ratio of the two different occupancies for the extracted element involved with the pattern having a length scale greater than the maximum length scale is an adjustment that involves adding a dummy wiring layer.

5. The method according to claim 1 , wherein step (6) of adjusting a mixing ratio of the two different occupancies for the extracted element involved with the pattern having a length scale greater than the maximum length scale is an adjustment of a wiring density of a wiring material in a wiring layer.

6. The method according to claim 1 , wherein step (6) of adjusting a mixing ratio of the two different occupancies for the extracted element involved with a pattern having a length scale greater than the maximum length scale is an adjustment of a wiring pattern in a wiring layer.

7. An information processing system for performing a calculation associated with controlling a warpage of a multilayered substrate, the information processing system comprising: a memory; a processing device communicatively coupled to the memory; and a calculation performance module communicatively coupled to the memory and processing device, wherein the calculation performance module is configured to perform a method comprising the steps of: (1) discretizing a two-dimensional surface of the multilayered substrate into finite planar elements, wherein the multilayered substrate has an upper wiring layer and a lower wiring layer, wherein the upper and lower wiring layers are made of at least two different materials having different elastic moduli, and wherein the materials are distributed two-dimensionally with different occupancies; (2) calculating a difference in coefficient of thermal expansion of the two-dimensional surface based on composite properties determined from two elastic moduli, two occupancies, and difference between the two occupancies obtained for each finite planar element; (3) calculating the warpage of the multilayered substrate based on the calculated composite properties of the two-dimensional surface according to a finite element method; (4) calculating a maximum length scale that corresponds to an allowable warpage; (5) extracting an element involved with a pattern having a length scale greater than a maximum length scale by digital filtering of the difference in coefficient of thermal expansion to produce an extracted element; (6) reducing the difference in coefficient of thermal expansion by adjusting a mixing ratio of the two different occupancies for the extracted element involved with a pattern having a length scale greater than the maximum length scale to produce an adjusted mixing ratio; (7) calculating the warpage of the multilayered substrate based on the adjusted mixing ratio of the two different occupancies according to the finite element method; and (8) repeating steps (5), (6) and (7) until the calculated warpage falls below a required warpage.

8. The system according to claim 7 , wherein step (5) of extracting an element involved with a pattern having a length scale greater than the maximum length scale by digital filtering of the difference in coefficient of thermal expansion is the digital filtering performed after a fast Fourier transform, and then an inverse fast Fourier transform is performed.